A synthesis framework for runoff prediction in ungauged basins - Runoff Prediction in Ungauged Basins: Synthesis across Processes, Places and Scales - page 22

Figure 2.10b . The runoff signature S Q (such as mean
annual runoff) is estimated from climate characteristics,
Cl, and catchment characteristics, Cc, based on a model f
(which can be a regression, a rainfall

Regression methods: In a regression method, the runoff
signature y of interest (for example, the flood discharge of

a given probability) is estimated from catchment and/or
climate characteristics x i with sampling error

^

runoff model, etc.).
The model will then differ between the regions, i.e., in
region 1, the S Q ¼

- ε :

X

p
i ¼ 1 β i x i þ ε þ η

f 2 (Cc,
Cl). Instead of Cc and Cl that were similar in the previous
approach, now it is f 1 , f 2 etc. that are similar within a group.
In some instances, the model f is process-based using
balance equations based on Newtonian mechanics. In other
instances the model f does not resolve the processes in
detail but exploits the co-evolution of catchments. For
example, even if the runoff processes are not known in
detail, a regression between stream network density and
mean annual floods can give excellent results. However,
the relationship between stream network density and mean
annual floods may differ fundamentally between regions.
In one region the low stream network density may be due
to karst, in another region the low stream network density
may be due to sandy soils, and in still another region it may
be due to low precipitation. In these three regions the
functional relationships, f, between stream network density
and mean annual floods will be different. Identifying
groups with similar regionalisation methods is less
straightforward than those with similar catchment charac-
teristics and, often, iterative methods are used.
Finally, the grouping of catchments is sometimes done on
the basis of runoff. This may be useful as a first regionalisa-
tion step. However, to transfer the runoff signatures to
ungauged catchments some sort of allocation rule is needed,
i.e., information about what group a particular ungauged
catchment belongs to. Allocation rules can, again, be esti-
mated from runoff data and then used for climate and
catchment characteristics in the ungauged catchments.

f 1 (Cc, Cl), in region 2, the S Q ¼

y

¼ β 0 þ

where there are i different characteristics, β i are the model

parameters (i.e., is the

model error. Many techniques are available to estimate
the model parameters for the linear model (e.g., Menden-
hall and Sincich, 2011 ). There are two options: use one
regression model for the entire domain of interest (termed
global regression); or subdivide the domain into regions
(according to Figure 2.10b ) and apply separate regression
models for each region (termed regional regression). From
a hydrological perspective it is important that the regres-
sion coefficients be interpreted hydrologically. This is
because such interpretations increase the likelihood that
the equation also applies to the ungauged catchments
that have not been used in estimating the coefficients.
The regression equations are very simple representations
of otherwise complex process relationships. These may, in
particular, involve co-evolutionary aspects of the catch-
ment. The interpretation of the coefficient is therefore not
necessarily mechanistic but may be based on a broader
reasoning of the co-evolution of landscapes, climate, soils
and vegetation.
Index methods: Index methods are based on some scaled
property of the catchment. For example, the flow duration
curve can be scaled by the mean annual flow. The index
method then assumes that the scaled flow duration curve is
uniform within a region as identified above. Another
example is the Budyko curve method, where the ratio of
mean annual actual evaporation to mean annual precipita-
tion is expressed as a function of the aridity index, the ratio
of mean annual potential evaporation and mean annual
precipitation. The index methods reflect some underlying
hydrological principle that is not inferred from the data but
from hydrological reasoning.
Geostatistical methods: The geostatistical methods
exploit the correlation of runoff signatures in space. In
the geostatistical approach the runoff signature of interest
in the ungauged catchment is assumed to be a weighted
mean of the runoff signatures in the neighbouring catch-
ments. The weights are estimated on the basis of the spatial
correlations of the runoff signatures and the relative loca-
tions of the catchments and/or the stream network. The
geostatistical approach goes beyond simple spatial distance
measures as they account for spatial correlations that will
differ between processes and regions (e.g., longer spatial
distances for low flows than for floods), and the so-called

regression coefficients) and η

2.3 From comparative hydrology to predictions
in ungauged basins
2.3.1 Statistical methods of predictions in ungauged
basins
There are two fundamentally different types of methods
available for estimating runoff in ungauged basins. The
first are statistical methods. In these methods the runoff
signatures of interest are assumed to be random variables.
Typically, the statistical methods are not based on balance
equations of mass, momentum and energy. Instead, they
consist of simple linear (or non-linear) relationships
between runoff, and climate and catchment characteristics.
The model structure is usually assumed a priori. The
model parameters, however, are usually estimated from
the data in the region of interest. In this topic, the statistical
methods have been assembled into three groups: